How do you find the roots of ##12x^3+31x^2-17x-6=0##?

Use the rational roots theorem, then factor to find:

##x = 2/3" "## or ##" "x = -1/4" "## or ##" "x = -3##

##f(x) = 12x^3+31x^2-17x-6##

By the rational roots theorem, any rational of ##f(x)## are expressible in the form ##p/q## for integers ##p, q## with ##p## a divisor of the constant term ##-6## and ##q## a divisor of the coefficient ##12## of the leading term.

That means that the only possible rational zeros are:

##+-1/12, +-1/6, +-1/4, +-1/3, +-1/2, +-2/3, +-3/4, +-1, +-3/2, +-2, +-3, +-6##

Notice that the signs of the coefficients of ##f(x)## are in the pattern ##+ + - -##. By Descartes' Rule of Signs, since there is one change of sign, this cubic has exactly one positive Real zero.

So let's look among the positive possibilities:

##f(1) = 12+31-17-6 = 20##

##f(1/2) = 12(1/8)+31(1/4)-17(1/2)-6 = (6+31-34-24)/4 = -21/4##

##f(2/3) = 12(8/27)+31(4/9)-17(2/3)-6 = (32+124-102-54)/9 = 0##

So ##x=2/3## is a zero and ##(3x-2)## a factor:

##12x^3+31x^2-17x-6 = (3x-2)(4x^2+13x+3)##

To factor the remaining quadratic, use an AC method:

Look for a pair of factors of ##AC=4*3=12## with sum ##B=13##.

The pair ##12, 1## works.

Use this pair to split the middle term and factor by grouping:

##4x^2+13x+3 = (4x^2+12x)+(x+3)##

##color(white)(4x^2+13x+3) = 4x(x+3)+1(x+3)##

##color(white)(4x^2+13x+3) = (4x+1)(x+3)##

So the other two zeros are:

##x = -1/4" "## and ##" "x = -3##